Category of modules

In algebra, given a ring R, the category of left modules over R is the category whose objects are all left modules over R and whose morphisms are all module homomorphisms between left R-modules. For example, when R is the ring of integers Z, it is the same thing as the category of abelian groups. The category of right modules is defined in a similar way.

Note: Some authors use the term module category for the category of modules. This term can be ambiguous since it could also refer to a category with a monoidal-category action.[1]

Properties

The category of left modules (or that of right modules) is an abelian category. The category has enough projectives[2] and enough injectives.[3] Mitchell's embedding theorem states every abelian category arises as a full subcategory of the category of modules.

Projective limits and inductive limits exist in the category of (say left) modules.[4]

Over a commutative ring, together with the tensor product of modules ⊗, the category of modules is a symmetric monoidal category.

Example: the category of vector spaces

The category K-Vect (some authors use VectK) has all vector spaces over a fixed field K as objects and K-linear transformations as morphisms. Since vector spaces over K (as a field) are the same thing as modules over the ring K, K-Vect is a special case of R-Mod, the category of left R-modules.

Much of linear algebra concerns the description of K-Vect. For example, the dimension theorem for vector spaces says that the isomorphism classes in K-Vect correspond exactly to the cardinal numbers, and that K-Vect is equivalent to the subcategory of K-Vect which has as its objects the free vector spaces Kn, where n is any cardinal number.

Generalizations

The category of sheaves of modules over a ringed space also has enough injectives (though not always enough projectives).

See also

References

  1. "module category in nLab". ncatlab.org.
  2. trivially since any module is a quotient of a free module.
  3. Dummit–Foote, Ch. 10, Theorem 38.
  4. Bourbaki, § 6.


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